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The paper surveys the known results concerning the question: “For what values of
n
does there exist a nonassociative Moufang loop of order
n
?” Proofs of the newest results for
n
odd, and a complete resolution of the case
n
even are also presented.
It has been proven by F. Leong and the first author (J. Algebra (1997), 474–486) that all Moufang loops of order
p
α
q
1
β
1
q
2
β
2
·
·
·
q
n
β
n
where
p
and
q
i
are odd primes, are associative if
p
<
q
1
<
q
2
<
·
·
·
<
q
n
, and
(i)
-
α
≤
3
\alpha \le 3
...
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